The billionaire faces a Friday deadline to complete the $44bn (£38bn) purchase, which he announced months ago but then tried to abandon.
He agreed to move forward earlier this month, just weeks before a court trial over those moves was due to start.
“Entering Twitter HQ – let that sink in!” Mr Musk wrote.
He also updated his bio to read “Chief Twit”.
Mr Musk has said the social media site needs significant changes. At least one report has suggested he is planning major job cuts.
When Elon Musk first revealed plans to buy Twitter, he said he wanted to clean up spam accounts on the platform and preserve it as a venue for free speech.
But Mr Musk, the world’s richest man and a prolific Twitter user known for his impulsive style, balked at the purchase just a few weeks later, citing concerns that the number of fake accounts on the platform was higher than Twitter claimed.
Twitter executives denied the accusations, arguing that Mr Musk wanted out because he was worried about the price.
It eventually filed a lawsuit to hold him to the deal, and Mr Musk revived his takeover plans for the company on the condition that legal proceedings were paused.
Investors voiced scepticism that the takeover would go ahead, especially as Mr Musk was seen as offering a heady price for a firm struggling to attract users and grow.
However, the deal must be completed by 28 October, or he will face trial over the contract.
“I’m excited about the Twitter situation,” Mr Musk said during a recent Tesla earnings call.
“I think it’s an asset that has just sort of languished for a long time but has incredible potential, although obviously myself and the other investors are overpaying for Twitter right now,” he added.
It came as Tesla, the electric car-maker, reportedly faces a criminal investigation in the US over claims that the company’s electric vehicles can drive themselves.
According to Reuters reports, three people familiar with the matter said that the US Department of Justice launched the investigation last year after more than a dozen crashes, some of them fatal, involving Tesla’s driver assistance system known as “Autopilot”, which was in use during the accidents.
During the same call, Mr Musk said Tesla would soon release an upgraded version of “full self-driving”, although its website says customers must keep their hands on the wheel and maintain control of their cars even when using the Autopilot feature.
Neither Twitter nor Tesla responded immediately to the BBC’s request for comment.